Zipf's law (original) (raw)

From Wikipedia, the free encyclopedia

Probability distribution

Zipf's Law on War and Peace.[1] The lower plot shows the remainder when the Zipf law is divided away. It shows that there remains significant pattern not fitted by Zipf law.

A plot of the frequency of each word as a function of its frequency rank for two English language texts: Culpeper's Complete Herbal (1652) and H. G. Wells's The War of the Worlds (1898) in a log-log scale. The dotted line is the ideal law y ∝ ⁠ 1 / x ⁠

Zipf's law (, German: [t͡sɪpf]) is an empirical law stating that when a list of measured values is sorted in decreasing order, the value of the n th entry is often approximately inversely proportional to n .

The best known instance of Zipf's law applies to the frequency table of words in a text or corpus of natural language: w o r d f r e q u e n c y ∝ 1 w o r d r a n k . {\displaystyle \ {\mathsf {word\ frequency}}\ \propto \ {\frac {1}{\ {\mathsf {word\ rank}}\ }}~.} {\displaystyle \ {\mathsf {word\ frequency}}\ \propto \ {\frac {1}{\ {\mathsf {word\ rank}}\ }}~.} It is usually found that the most common word occurs approximately twice as often as the next common one, three times as often as the third most common, and so on. For example, in the Brown Corpus of American English text, the word "the" is the most frequently occurring word, and by itself accounts for nearly 7% of all word occurrences (69,971 out of slightly over 1 million). True to Zipf's law, the second-place word "of" accounts for slightly over 3.5% of words (36,411 occurrences), followed by "and" (28,852).[2] It is often used in the following form, called Zipf-Mandelbrot law: f r e q u e n c y ∝ 1 ( r a n k + b ) a {\displaystyle \ {\mathsf {frequency}}\ \propto \ {\frac {1}{\ \left(\ {\mathsf {rank}}+b\ \right)^{a}\ }}\ } {\displaystyle \ {\mathsf {frequency}}\ \propto \ {\frac {1}{\ \left(\ {\mathsf {rank}}+b\ \right)^{a}\ }}\ } where a {\displaystyle \ a\ } {\displaystyle \ a\ } and b {\displaystyle \ b\ } {\displaystyle \ b\ } are fitted parameters, with a ≈ 1 , {\displaystyle \ a\approx 1\ ,} {\displaystyle \ a\approx 1\ ,} and b ≈ 2.7 . {\displaystyle \ b\approx 2.7~.} {\displaystyle \ b\approx 2.7~.}[1]

This law is named after the American linguist George Kingsley Zipf,[3][4][5] and is still an important concept in quantitative linguistics. It has been found to apply to many other types of data studied in the physical and social sciences.

In mathematical statistics, the concept has been formalized as the Zipfian distribution: A family of related discrete probability distributions whose rank-frequency distribution is an inverse power law relation. They are related to Benford's law and the Pareto distribution.

Some sets of time-dependent empirical data deviate somewhat from Zipf's law. Such empirical distributions are said to be quasi-Zipfian.

In 1913, the German physicist Felix Auerbach observed an inverse proportionality between the population sizes of cities, and their ranks when sorted by decreasing order of that variable.[6]

Zipf's law had been discovered before Zipf,[a] first by the French stenographer Jean-Baptiste Estoup in 1916,[8][7] and also by G. Dewey in 1923,[9] and by E. Condon in 1928.[10]

The same relation for frequencies of words in natural language texts was observed by George Zipf in 1932,[4] but he never claimed to have originated it. In fact, Zipf did not like mathematics. In his 1932 publication,[11] the author speaks with disdain about mathematical involvement in linguistics, a.o. ibidem, p. 21:

... let me say here for the sake of any mathematician who may plan to formulate the ensuing data more exactly, the ability of the highly intense positive to become the highly intense negative, in my opinion, introduces the devil into the formula in the form of − i . {\displaystyle \ {\sqrt {-i\;}}~.} {\displaystyle \ {\sqrt {-i\;}}~.}

The only mathematical expression Zipf used looks like a._b_2 = constant, which he "borrowed" from Alfred J. Lotka's 1926 publication.[12]

The same relationship was found to occur in many other contexts, and for other variables besides frequency.[1] For example, when corporations are ranked by decreasing size, their sizes are found to be inversely proportional to the rank.[13] The same relation is found for personal incomes (where it is called Pareto principle[14]), number of people watching the same TV channel,[15] notes in music,[16] cells transcriptomes,[17][18] and more.

In 1992 bioinformatician Wentian Li published a short paper[19] showing that Zipf's law emerges even in randomly generated texts. It included proof that the power law form of Zipf's law was a byproduct of ordering words by rank.

Zipf's law

Probability mass functionPlot of the Zipf PMF for N = 10 .Zipf PMF for N = 10 on a log–log scale. The horizontal axis is the index k . (The function is only defined at integer values of k . The connecting lines are only visual guides; they do not indicate continuity.)
Cumulative distribution functionPlot of the Zipf CDF for N=10Zipf CDF for N = 10 . The horizontal axis is the index k . (The function is only defined at integer values of k . The connecting lines do not indicate continuity.)
Parameters s ≥ 0 {\displaystyle s\geq 0\,} {\displaystyle s\geq 0\,} (real) N ∈ { 1 , 2 , 3 … } {\displaystyle N\in \{1,2,3\ldots \}} {\displaystyle N\in \{1,2,3\ldots \}} (integer)
Support k ∈ { 1 , 2 , … , N } {\displaystyle k\in \{1,2,\ldots ,N\}} {\displaystyle k\in \{1,2,\ldots ,N\}}
PMF 1 / k s H N , s {\displaystyle {\frac {1/k^{s}}{H_{N,s}}}} {\displaystyle {\frac {1/k^{s}}{H_{N,s}}}} where HN,s is the _N_th generalized harmonic number
CDF H k , s H N , s {\displaystyle {\frac {H_{k,s}}{H_{N,s}}}} {\displaystyle {\frac {H_{k,s}}{H_{N,s}}}}
Mean H N , s − 1 H N , s {\displaystyle {\frac {H_{N,s-1}}{H_{N,s}}}} {\displaystyle {\frac {H_{N,s-1}}{H_{N,s}}}}
Mode 1 {\displaystyle 1\,} {\displaystyle 1\,}
Variance H N , s − 2 H N , s − H N , s − 1 2 H N , s 2 {\displaystyle {\frac {H_{N,s-2}}{H_{N,s}}}-{\frac {H_{N,s-1}^{2}}{H_{N,s}^{2}}}} {\displaystyle {\frac {H_{N,s-2}}{H_{N,s}}}-{\frac {H_{N,s-1}^{2}}{H_{N,s}^{2}}}}
Entropy s H N , s ∑ k = 1 N ln ⁡ ( k ) k s + ln ⁡ ( H N , s ) {\displaystyle {\frac {s}{H_{N,s}}}\sum \limits _{k=1}^{N}{\frac {\ln(k)}{k^{s}}}+\ln(H_{N,s})} {\displaystyle {\frac {s}{H_{N,s}}}\sum \limits _{k=1}^{N}{\frac {\ln(k)}{k^{s}}}+\ln(H_{N,s})}
MGF 1 H N , s ∑ n = 1 N e n t n s {\displaystyle {\frac {1}{H_{N,s}}}\sum \limits _{n=1}^{N}{\frac {e^{nt}}{n^{s}}}} {\displaystyle {\frac {1}{H_{N,s}}}\sum \limits _{n=1}^{N}{\frac {e^{nt}}{n^{s}}}}
CF 1 H N , s ∑ n = 1 N e i n t n s {\displaystyle {\frac {1}{H_{N,s}}}\sum \limits _{n=1}^{N}{\frac {e^{int}}{n^{s}}}} {\displaystyle {\frac {1}{H_{N,s}}}\sum \limits _{n=1}^{N}{\frac {e^{int}}{n^{s}}}}

Formally, the Zipf distribution on N elements assigns to the element of rank k (counting from 1) the probability

f ( k ; N ) = { 1 H N 1 k , if 1 ≤ k ≤ N , 0 , if k < 1 or N < k . {\displaystyle \ f(k;N)~=~{\begin{cases}{\frac {1}{\ H_{N}}}\ {\frac {1}{\ k\ }}\ ,&\ {\mbox{ if }}\ 1\leq k\leq N~,\\{}\\~~0~~,&\ {\mbox{ if }}\ k<1\ {\mbox{ or }}\ N<k~.\end{cases}}} {\displaystyle \ f(k;N)~=~{\begin{cases}{\frac {1}{\ H_{N}}}\ {\frac {1}{\ k\ }}\ ,&\ {\mbox{ if }}\ 1\leq k\leq N~,\\{}\\~~0~~,&\ {\mbox{ if }}\ k<1\ {\mbox{ or }}\ N<k~.\end{cases}}}

where HN is a normalization constant: The Nth harmonic number:

H N ≡ ∑ k = 1 N 1 k . {\displaystyle H_{N}\equiv \sum _{k=1}^{N}{\frac {\ 1\ }{k}}~.} {\displaystyle H_{N}\equiv \sum _{k=1}^{N}{\frac {\ 1\ }{k}}~.}

The distribution is sometimes generalized to an inverse power law with exponent s instead of 1 .[20] Namely,

f ( k ; N , s ) = 1 H N , s 1 k s {\displaystyle f(k;N,s)={\frac {1}{H_{N,s}}}\,{\frac {1}{k^{s}}}} {\displaystyle f(k;N,s)={\frac {1}{H_{N,s}}}\,{\frac {1}{k^{s}}}}

where HN,s is a generalized harmonic number

H N , s = ∑ k = 1 N 1 k s . {\displaystyle H_{N,s}=\sum _{k=1}^{N}{\frac {1}{k^{s}}}~.} {\displaystyle H_{N,s}=\sum _{k=1}^{N}{\frac {1}{k^{s}}}~.}

The generalized Zipf distribution can be extended to infinitely many items (N = ∞) only if the exponent s exceeds 1 . In that case, the normalization constant HN,s becomes Riemann's zeta function,

ζ ( s ) = ∑ k = 1 ∞ 1 k s < ∞ . {\displaystyle \zeta (s)=\sum _{k=1}^{\infty }{\frac {1}{k^{s}}}<\infty ~.} {\displaystyle \zeta (s)=\sum _{k=1}^{\infty }{\frac {1}{k^{s}}}<\infty ~.}

The infinite item case is characterized by the Zeta distribution and is called Lotka's law. If the exponent s is 1 or less, the normalization constant HN,s diverges as N tends to infinity.

Empirically, a data set can be tested to see whether Zipf's law applies by checking the goodness of fit of an empirical distribution to the hypothesized power law distribution with a Kolmogorov–Smirnov test, and then comparing the (log) likelihood ratio of the power law distribution to alternative distributions like an exponential distribution or lognormal distribution.[21]

Zipf's law can be visuallized by plotting the item frequency data on a log-log graph, with the axes being the logarithm of rank order, and logarithm of frequency. The data conform to Zipf's law with exponent s to the extent that the plot approximates a linear (more precisely, affine) function with slope −s. For exponent s = 1 , one can also plot the reciprocal of the frequency (mean interword interval) against rank, or the reciprocal of rank against frequency, and compare the result with the line through the origin with slope 1 .[3]

Statistical explanations

[edit]

Although Zipf's Law holds for most natural languages, and even some non-natural ones like Esperanto[22] and Toki Pona,[23] the reason is still not well understood.[24] Recent reviews of generative processes for Zipf's law include Mitzenmacher, "A Brief History of Generative Models for Power Law and Lognormal Distributions",[25] and Simkin, "Re-inventing Willis".[26]

However, it may be partly explained by statistical analysis of randomly generated texts. Wentian Li has shown that in a document in which each character has been chosen randomly from a uniform distribution of all letters (plus a space character), the "words" with different lengths follow the macro-trend of Zipf's law (the more probable words are the shortest and have equal probability).[27] In 1959, Vitold Belevitch observed that if any of a large class of well-behaved statistical distributions (not only the normal distribution) is expressed in terms of rank and expanded into a Taylor series, the first-order truncation of the series results in Zipf's law. Further, a second-order truncation of the Taylor series resulted in Mandelbrot's law.[28][29]

The principle of least effort is another possible explanation: Zipf himself proposed that neither speakers nor hearers using a given language wants to work any harder than necessary to reach understanding, and the process that results in approximately equal distribution of effort leads to the observed Zipf distribution.[5][30]

A minimal explanation assumes that words are generated by monkeys typing randomly. If language is generated by a single monkey typing randomly, with fixed and nonzero probability of hitting each letter key or white space, then the words (letter strings separated by white spaces) produced by the monkey follows Zipf's law.[31]

Another possible cause for the Zipf distribution is a preferential attachment process, in which the value x of an item tends to grow at a rate proportional to x (intuitively, "the rich get richer" or "success breeds success"). Such a growth process results in the Yule–Simon distribution, which has been shown to fit word frequency versus rank in language[32] and population versus city rank[33] better than Zipf's law. It was originally derived to explain population versus rank in species by Yule, and applied to cities by Simon.

A similar explanation is based on atlas models, systems of exchangeable positive-valued diffusion processes with drift and variance parameters that depend only on the rank of the process. It has been shown mathematically that Zipf's law holds for Atlas models that satisfy certain natural regularity conditions.[34][35]

A generalization of Zipf's law is the Zipf–Mandelbrot law, proposed by Benoit Mandelbrot, whose frequencies are:

f ( k ; N , q , s ) = 1 C 1 ( k + q ) s . {\displaystyle f(k;N,q,s)={\frac {1}{\ C\ }}\ {\frac {1}{\ \left(k+q\right)^{s}}}~.} {\displaystyle f(k;N,q,s)={\frac {1}{\ C\ }}\ {\frac {1}{\ \left(k+q\right)^{s}}}~.}[_clarification needed_]

The constant C is the Hurwitz zeta function evaluated at s.

Zipfian distributions can be obtained from Pareto distributions by an exchange of variables.[20]

The Zipf distribution is sometimes called the discrete Pareto distribution[36] because it is analogous to the continuous Pareto distribution in the same way that the discrete uniform distribution is analogous to the continuous uniform distribution.

The tail frequencies of the Yule–Simon distribution are approximately

f ( k ; ρ ) ≈ [ c o n s t a n t ] k ( ρ + 1 ) {\displaystyle f(k;\rho )\approx {\frac {\ [{\mathsf {constant}}]\ }{k^{(\rho +1)}}}} {\displaystyle f(k;\rho )\approx {\frac {\ [{\mathsf {constant}}]\ }{k^{(\rho +1)}}}}

for any choice of ρ > 0 .

In the parabolic fractal distribution, the logarithm of the frequency is a quadratic polynomial of the logarithm of the rank. This can markedly improve the fit over a simple power-law relationship.[37] Like fractal dimension, it is possible to calculate Zipf dimension, which is a useful parameter in the analysis of texts.[38]

It has been argued that Benford's law is a special bounded case of Zipf's law,[37] with the connection between these two laws being explained by their both originating from scale invariant functional relations from statistical physics and critical phenomena.[39] The ratios of probabilities in Benford's law are not constant. The leading digits of data satisfying Zipf's law with s = 1 , satisfy Benford's law.

n {\displaystyle n} {\displaystyle n} Benford's law: P ( n ) = {\displaystyle P(n)=} {\displaystyle P(n)=} log 10 ⁡ ( n + 1 ) − log 10 ⁡ ( n ) {\displaystyle \log _{10}(n+1)-\log _{10}(n)} {\displaystyle \log _{10}(n+1)-\log _{10}(n)} log ⁡ ( P ( n ) / P ( n − 1 ) ) log ⁡ ( n / ( n − 1 ) ) {\displaystyle {\frac {\log(P(n)/P(n-1))}{\log(n/(n-1))}}} {\displaystyle {\frac {\log(P(n)/P(n-1))}{\log(n/(n-1))}}}
1 0.30103000
2 0.17609126 −0.7735840
3 0.12493874 −0.8463832
4 0.09691001 −0.8830605
5 0.07918125 −0.9054412
6 0.06694679 −0.9205788
7 0.05799195 −0.9315169
8 0.05115252 −0.9397966
9 0.04575749 −0.9462848

Following Auerbach's 1913 observation, there has been substantial examination of Zipf's law for city sizes.[40] However, more recent empirical[41][42] and theoretical[43] studies have challenged the relevance of Zipf's law for cities.

Word frequencies in natural languages

[edit]

Zipf's law plot for the first 10 million words in 30 Wikipedias (as of October 2015) in a log-log scale

In many texts in human languages, word frequencies approximately follow a Zipf distribution with exponent s close to 1 ; that is, the most common word occurs about n times the nth most common one.

The actual rank-frequency plot of a natural language text deviates in some extent from the ideal Zipf distribution, especially at the two ends of the range. The deviations may depend on the language, on the topic of the text, on the author, on whether the text was translated from another language, and on the spelling rules used.[_citation needed_] Some deviation is inevitable because of sampling error.

At the low-frequency end, where the rank approaches N, the plot takes a staircase shape, because each word can occur only an integer number of times.

A log-log plot of word frequency in the English Wikipedia (27 November 2006). 'Most popular words are "the", "of" and "and", as expected. Zipf's law corresponds to the middle linear portion of the curve, roughly following the green (⁠1/_x_⁠) line, while the early part is closer to the magenta (⁠1/√_x_⁠) line while the later part is closer to the cyan (⁠1/ _x_2 ⁠) line. These lines correspond to three distinct parameterizations of the Zipf–Mandelbrot distribution, overall a broken power law with three segments: a head, middle, and tail.[_citation needed_] Other descriptions highlight two segments or "regimes" instead.[44][45]

.

In some Romance languages, the frequencies of the dozen or so most frequent words deviate significantly from the ideal Zipf distribution, because of those words include articles inflected for grammatical gender and number.[_citation needed_]

In many East Asian languages, such as Chinese, Lhasa Tibetan, and Vietnamese, each "word" consists of a single syllable; a word of English being often translated to a compound of two such syllables. The rank-frequency table for those "words" deviates significantly from the ideal Zipf law, at both ends of the range.[_citation needed_]

Even in English, the deviations from the ideal Zipf's law become more apparent as one examines large collections of texts. Analysis of a corpus of 30,000 English texts showed that only about 15% of the texts in it have a good fit to Zipf's law. Slight changes in the definition of Zipf's law can increase this percentage up to close to 50%.[46]

In these cases, the observed frequency-rank relation can be modeled more accurately as by separate Zipf–Mandelbrot laws distributions for different subsets or subtypes of words. This is the case for the frequency-rank plot of the first 10 million words of the English Wikipedia. In particular, the frequencies of the closed class of function words in English is better described with s lower than 1 , while open-ended vocabulary growth with document size and corpus size require s greater than 1 for convergence of the Generalized Harmonic Series.[3]

Well's War of the Worlds in plain text, in a book code, and in a Vigenère cipher

When a text is encrypted in such a way that every occurrence of each distinct plaintext word is always mapped to the same encrypted word (as in the case of simple substitution ciphers, like the Caesar ciphers, or simple codebook ciphers), the frequency-rank distribution is not affected. On the other hand, if separate occurrences of the same word may be mapped to two or more different words (as happens with the Vigenère cipher), the Zipf distribution will typically have a flat part at the high-frequency end.[_citation needed_]

Zipf's law has been used for extraction of parallel fragments of texts out of comparable corpora.[47] Laurance Doyle and others have suggested the application of Zipf's law for detection of alien language in the search for extraterrestrial intelligence.[48][49]

The frequency-rank word distribution is often characteristic of the author and changes little over time. This feature has been used in the analysis of texts for authorship attribution.[50][51]

The word-like sign groups of the 15th-century codex Voynich Manuscript have been found to satisfy Zipf's law, suggesting that text is most likely not a hoax but rather written in an obscure language or cipher.[52][53]

  1. ^ as Zipf acknowledged[5]: 546

  2. ^ a b c Piantadosi, Steven (25 March 2014). "Zipf's word frequency law in natural language: A critical review and future directions". Psychon Bull Rev. 21 (5): 1112–1130. doi:10.3758/s13423-014-0585-6. PMC 4176592. PMID 24664880.

  3. ^ Fagan, Stephen; Gençay, Ramazan (2010). "An introduction to textual econometrics". In Ullah, Aman; Giles, David E.A. (eds.). Handbook of Empirical Economics and Finance. CRC Press. pp. 133–153, esp.&nbps, 139. ISBN 9781420070361. For example, in the Brown Corpus, consisting of over one million words, half of the word volume consists of repeated uses of only 135 words.

  4. ^ a b c Powers, David M.W. (1998). Applications and explanations of Zipf's law. Joint conference on new methods in language processing and computational natural language learning. Association for Computational Linguistics. pp. 151–160. Archived from the original on 10 September 2015. Retrieved 2 February 2015 – via aclweb.org.

  5. ^ a b Zipf, G.K. (1935). The Psychobiology of Language. New York, NY: Houghton-Mifflin.

  6. ^ a b c Zipf, George K. (1949). Human Behavior and the Principle of Least Effort. Cambridge, MA: Addison-Wesley. p. 1 – via archive.org.

  7. ^ Auerbach, F. (1913). "Das Gesetz der Bevölkerungskonzentration". Petermann's Geographische Mitteilungen (in German). 59: 74–76.

  8. ^ a b Manning, Christopher D.; Schütze, Hinrich (1999). Foundations of Statistical Natural Language Processing. MIT Press. p. 24. ISBN 978-0-262-13360-9.

  9. ^ Estoup, J.-B. (1916). Gammes Stenographiques (4th ed.). Cited in Manning & Schütze (1999).[7]

  10. ^ Dewey, Godfrey (1923). Relative Frequency of English Speech Sounds. Harvard University Press – via Internet Archive.

  11. ^ Condon, E.U. (1928). "Statistics of vocabulary". Science. 67 (1733): 300. Bibcode:1928Sci....67..300C. doi:10.1126/science.67.1733.300. PMID 17782935.

  12. ^ Zipf, G.K. (1932). Selected Studies on the Principle of Relative Frequency in Language. Harvard, MA: Harvard University Press.

  13. ^ Zipf, George Kingsley (1942). "The unity of nature, least-action, and natural social science". Sociometry. 5 (1): 48–62. doi:10.2307/2784953. ISSN 0038-0431. JSTOR 2784953. Archived from the original on 20 November 2022. Retrieved 20 November 2022.

  14. ^ Axtell, Robert L. (2001). "Zipf distribution of US firm sizes". Science. 293 (5536): 1818–1820. Bibcode:2001Sci...293.1818A. doi:10.1126/science.1062081. PMID 11546870. Archived from the original on 17 October 2023 – via science.org.

  15. ^ Sandmo, Agnar (January 2015). "Chapter 1 - The principal problem in political economy: Income distribution in the history of economic thought". In Atkinson, Anthony B.; Bourguignon, François (eds.). Handbook of Income Distribution. Vol. 2. Elsevier. pp. 3–65. doi:10.1016/B978-0-444-59428-0.00002-3. ISBN 978-0-444-59430-3. Archived from the original on 29 October 2023. Retrieved 11 July 2023 – via Elsevier (sciencedirect.com).

  16. ^ M. Eriksson, S.M. Hasibur Rahman, F. Fraille, M. Sjöström, Efficient Interactive Multicast over DVB-T2 - Utilizing Dynamic SFNs and PARPS Archived 2014-05-02 at the Wayback Machine, 2013 IEEE International Conference on Computer and Information Technology (BMSB'13), London, UK, June 2013. Suggests a heterogeneous Zipf-law TV channel-selection model

  17. ^ Zanette, Damián H. (7 June 2004). "Zipf's law and the creation of musical context". arXiv:cs/0406015.

  18. ^ Lazzardi, Silvia; Valle, Filippo; Mazzolini, Andrea; Scialdone, Antonio; Caselle, Michele; Osella, Matteo (17 June 2021). "Emergent statistical laws in single-cell transcriptomic data". bioRxiv: 2021–06.16.448706. doi:10.1101/2021.06.16.448706. S2CID 235482777. Archived from the original on 17 June 2021. Retrieved 18 June 2021.

  19. ^ Chenna, Ramu; Gibson, Toby (2011). Evaluation of the suitability of a Zipfian gap model for pairwise sequence alignment (PDF). International Conference on Bioinformatics Computational Biology. BIC 4329. Archived from the original (PDF) on 6 March 2014.

  20. ^ Li, Wentian (1992). "Random texts exhibit Zipf's-law-like word frequency distribution". IEEE Transactions on Information Theory. 38 (6): 1842–1845. doi:10.1109/18.165464 – via IEEE Xplore.

  21. ^ a b Adamic, Lada A. (2000). Zipf, power-laws, and Pareto – a ranking tutorial (Report) (re-issue ed.). Hewlett-Packard Company. Archived from the original on 1 April 2023. Retrieved 12 October 2023. "original publication". www.parc.xerox.com. Xerox Corporation. Archived from the original on 7 November 2001. Retrieved 23 February 2016.

  22. ^ Clauset, A.; Shalizi, C.R.; Newman, M.E.J. (2009). "Power-law distributions in empirical data". SIAM Review. 51 (4): 661–703. arXiv:0706.1062. Bibcode:2009SIAMR..51..661C. doi:10.1137/070710111.

  23. ^ Manaris, Bill; Pellicoro, Luca; Pothering, George; Hodges, Harland (13 February 2006). Investigating Esperanto's statistical proportions relative to other languages using neural networks and Zipf's law (PDF). Artificial Intelligence and Applications. Innsbruck, Austria. pp. 102–108. Archived from the original (PDF) on 5 March 2016 – via cs.cofc.edu.

  24. ^ Skotarek, Dariusz (12–14 October 2020). Zipf's law in Toki Pona (PDF). ExLing 2020: 11th International Conference of Experimental Linguistics. Athens, Greece: ExLing Society. doi:10.36505/ExLing-2020/11/0047/000462. ISBN 978-618-84585-1-2 – via exlingsociety.com.

  25. ^ Brillouin, Léon (2004) [1959, 1988]. La science et la théorie de l'information [_The Science and Theory of Information_] (in French). réédité en 1988, traduction anglaise rééditée en 2004

  26. ^ Mitzenmacher, Michael (January 2004). "A brief history of generative models for power law and lognormal distributions". Internet Mathematics. 1 (2): 226–251. doi:10.1080/15427951.2004.10129088. ISSN 1542-7951. S2CID 1671059. Archived from the original on 22 July 2023. Retrieved 25 July 2023.

  27. ^ Simkin, M.V.; Roychowdhury, V.P. (May 2011). "Re-inventing Willis". Physics Reports. 502 (1): 1–35. arXiv:physics/0601192. Bibcode:2011PhR...502....1S. doi:10.1016/j.physrep.2010.12.004. ISSN 0370-1573. S2CID 88517297. Archived from the original on 29 January 2012. Retrieved 25 July 2023.

  28. ^ Li, Wentian (1992). "Random Texts Exhibit Zipf's-Law-Like Word Frequency Distribution". IEEE Transactions on Information Theory. 38 (6): 1842–1845. CiteSeerX 10.1.1.164.8422. doi:10.1109/18.165464.

  29. ^ Belevitch, V. (18 December 1959). "On the statistical laws of linguistic distributions" (PDF). Annales de la Société Scientifique de Bruxelles. 73: 310–326. Archived (PDF) from the original on 15 December 2020. Retrieved 24 April 2020.

  30. ^ Neumann, P.G. (c. 2022). Statistical metalinguistics and Zipf / Pareto / Mandelbrot (Report). Computer Science Laboratory. Vol. 12A. Menlo Park, CA: SRI International. Archived from the original on 5 June 2011. Retrieved 29 May 2011 – via sri.com.

  31. ^ Ferrer i Cancho, Ramon & Sole, Ricard V. (2003). "Least effort and the origins of scaling in human language". Proceedings of the National Academy of Sciences of the United States of America. 100 (3): 788–791. Bibcode:2003PNAS..100..788C. doi:10.1073/pnas.0335980100. PMC 298679. PMID 12540826.

  32. ^ Conrad, B.; Mitzenmacher, M. (July 2004). "Power laws for monkeys typing randomly: The case of unequal probabilities". IEEE Transactions on Information Theory. 50 (7): 1403–1414. doi:10.1109/TIT.2004.830752. ISSN 1557-9654. S2CID 8913575. Archived from the original on 17 October 2022. Retrieved 20 August 2023.

  33. ^ Lin, Ruokuang; Ma, Qianli D.Y.; Bian, Chunhua (2014). "Scaling laws in human speech, decreasing emergence of new words, and a generalized model". arXiv:1412.4846 [cs.CL].

  34. ^ Vitanov, Nikolay K.; Ausloos, Marcel; Bian, Chunhua (2015). "Test of two hypotheses explaining the size of populations in a system of cities". Journal of Applied Statistics. 42 (12): 2686–2693. arXiv:1506.08535. Bibcode:2015JApSt..42.2686V. doi:10.1080/02664763.2015.1047744. S2CID 10599428.

  35. ^ Ricardo T. Fernholz; Robert Fernholz (December 2020). "Zipf's law for atlas models". Journal of Applied Probability. 57 (4): 1276–1297. doi:10.1017/jpr.2020.64. S2CID 146808080. Archived from the original on 29 January 2021. Retrieved 26 March 2021.

  36. ^ Terence Tao (2012). "E Pluribus Unum: From Complexity, Universality". Daedalus. 141 (3): 23–34. doi:10.1162/DAED_a_00158. S2CID 14535989. Archived from the original on 5 August 2021. Retrieved 26 March 2021.

  37. ^ Johnson, N.L.; Kotz, S. & Kemp, A. W. (1992). Univariate Discrete Distributions (second ed.). New York: John Wiley & Sons, Inc. p. 466. ISBN 978-0-471-54897-3.

  38. ^ a b van der Galien, Johan Gerard (8 November 2003). "Factorial randomness: The laws of Benford and Zipf with respect to the first digit distribution of the factor sequence from the natural numbers". zonnet.nl. Archived from the original on 5 March 2007. Retrieved 8 July 2016.

  39. ^ Eftekhari, Ali (2006). "Fractal geometry of texts: An initial application to the works of Shakespeare". Journal of Quantitative Linguistic. 13 (2–3): 177–193. doi:10.1080/09296170600850106. S2CID 17657731.

  40. ^ Pietronero, L.; Tosatti, E.; Tosatti, V.; Vespignani, A. (2001). "Explaining the uneven distribution of numbers in nature: The laws of Benford and Zipf". Physica A. 293 (1–2): 297–304. arXiv:cond-mat/9808305. Bibcode:2001PhyA..293..297P. doi:10.1016/S0378-4371(00)00633-6.

  41. ^ Gabaix, Xavier (1999). "Zipf's law for cities: An explanation". The Quarterly Journal of Economics. 114 (3): 739–767. doi:10.1162/003355399556133. ISSN 0033-5533. JSTOR 2586883. Archived from the original on 26 October 2021. Retrieved 26 October 2021 – via jstor.org.

  42. ^ Arshad, Sidra; Hu, Shougeng; Ashraf, Badar Nadeem (15 February 2018). "Zipf's law and city size distribution: A survey of the literature and future research agenda". Physica A: Statistical Mechanics and Its Applications. 492: 75–92. Bibcode:2018PhyA..492...75A. doi:10.1016/j.physa.2017.10.005. ISSN 0378-4371. Archived from the original on 29 October 2023. Retrieved 26 October 2021.

  43. ^ Gan, Li; Li, Dong; Song, Shunfeng (1 August 2006). "Is the Zipf law spurious in explaining city-size distributions?". Economics Letters. 92 (2): 256–262. doi:10.1016/j.econlet.2006.03.004. ISSN 0165-1765. Archived from the original on 13 April 2019. Retrieved 26 October 2021.

  44. ^ Verbavatz, Vincent; Barthelemy, Marc (November 2020). "The growth equation of cities". Nature. 587 (7834): 397–401. arXiv:2011.09403. Bibcode:2020Natur.587..397V. doi:10.1038/s41586-020-2900-x. ISSN 1476-4687. PMID 33208958. S2CID 227012701. Archived from the original on 29 October 2021. Retrieved 26 October 2021.

  45. ^ Ferrer Cancho, Ramon; Solé, Ricard V. (December 2001). "Two Regimes in the Frequency of Words and the Origins of Complex Lexicons: Zipf's Law Revisited". Journal of Quantitative Linguistics. 8 (3): 165–173. doi:10.1076/jqul.8.3.165.4101. hdl:2117/180381. ISSN 0929-6174.

  46. ^ Dorogovtsev, S. N.; Mendes, J. F. F. (22 December 2001). "Language as an evolving word web". Proceedings of the Royal Society of London. Series B: Biological Sciences. 268 (1485): 2603–2606. doi:10.1098/rspb.2001.1824. ISSN 0962-8452. PMC 1088922. PMID 11749717.

  47. ^ Moreno-Sánchez, I.; Font-Clos, F.; Corral, A. (2016). "Large-scale analysis of Zipf's Law in English texts". PLOS ONE. 11 (1): e0147073. arXiv:1509.04486. Bibcode:2016PLoSO..1147073M. doi:10.1371/journal.pone.0147073. PMC 4723055. PMID 26800025.

  48. ^ Mohammadi, Mehdi (2016). "Parallel Document Identification using Zipf's Law" (PDF). Proceedings of the Ninth Workshop on Building and Using Comparable Corpora. LREC 2016. Portorož, Slovenia. pp. 21–25. Archived (PDF) from the original on 23 March 2018.

  49. ^ Doyle, L.R. (18 November 2016). "Why alien language would stand out among all the noise of the universe". Nautilus Quarterly. Archived from the original on 29 July 2020. Retrieved 30 August 2020.

  50. ^ Kershenbaum, Arik (16 March 2021). The Zoologist's Guide to the Galaxy: What animals on Earth reveal about aliens – and ourselves. Penguin. pp. 251–256. ISBN 978-1-9848-8197-7. OCLC 1242873084.

  51. ^ van Droogenbroeck, Frans J. (2016). Handling the Zipf distribution in computerized authorship attribution (Report). Archived from the original on 4 October 2023 – via academia.edu.

  52. ^ van Droogenbroeck, Frans J. (2019). An essential rephrasing of the Zipf-Mandelbrot law to solve authorship attribution applications by Gaussian statistics (Report). Archived from the original on 30 September 2023 – via academia.edu.

  53. ^ Boyle, Rebecca. "Mystery text's language-like patterns may be an elaborate hoax". New Scientist. Archived from the original on 18 May 2022. Retrieved 25 February 2022.

  54. ^ Montemurro, Marcelo A.; Zanette, Damián H. (21 June 2013). "Keywords and co-occurrence patterns in the Voynich manuscript: An information-theoretic analysis". PLoS One. 8 (6): e66344. Bibcode:2013PLoSO...866344M. doi:10.1371/journal.pone.0066344. ISSN 1932-6203. PMC 3689824. PMID 23805215.

Wikimedia Commons has media related to Zipf's law.